Binary Data Analysis of Randomized Clinical Trials with Noncompliance by Lui Kung-Jong;

Author:Lui, Kung-Jong;
Language: eng
Format: epub
ISBN: 4041506
Publisher: Wiley
Published: 2011-03-20T16:00:00+00:00

4.1 Testing superiority

Recall that under the monotonicity and exclusion restriction assumptions (for always-takers and never-takers), the ITT analysis estimates the PD, =, where , the difference in the conditional probabilities of a positive response among compliers between two treatments (Chapter 2). Since = 0 if and only if Δ = 0. Thus, to test the null hypothesis H0: Δ = 0, we may apply the ITT analysis to test H0: δ = 0. Define

(4.5)

where (or ) actually represents the proportion of a positive response among patients assigned to treatment g (= 1 for experimental, and = 0 for standard) despite what treatment they actually receive. Define . One can show that the expectation under the above assumptions is (Exercise 4.7):

(4.6)

We can further show that the variance for is given by

(4.7)

Following Light and Margolin (1971) and Rae (1988), we can estimate the intraclass correlation ρg by (Exercise 4.8):

(4.8)

where , the summation of S is over CA and N for g = 1, and is over CN and A for g = 0. The discussion on interval estimation of the intraclass correlation under the Dirichlet-multinomial distribution appears elsewhere (Lui, Cumberland and Mayer et al., 1999). On the basis of (4.8), we can estimate the variance under H0 :Δ = 0 by

(4.9)

where is the pooled estimate of the two proportions of a positive response for the two treatments. On the basis of (4.9), we may consider the test statistic

(4.10)

We will reject H0 :Δ = 0 at the α-level if the test statistic (4.10), or , where Zα is the upper 100(α)th percentile of the standard normal distribution. Although our interest is to detect the superiority of the experimental treatment to the standard treatment, as noted in previous chapters, we may wish to do a two-sided test rather than a one-sided test for ethical and safety reasons (Fleiss, 1981). We will claim that the experimental treatment is superior to the standard treatment if the test statistic (4.10), holds. Note also that in application of statistic (4.10) for testing superiority, we may also consider using tanh−1(x)( = 0.5log((1+x)/(1−x))) transformation (Edwardes, 1995; Lui, 2002). Since the asymptotic variance under H0 :Δ = 0 is the same as , we obtain the following test statistic (Lui and Chang, 2011a),

(4.11)

where is given in (4.9). If the test statistic (4.11), or , we will reject H0 :Δ = 0 at the α- level. In particular, we will claim that the experimental treatment is superior to the standard treatment if the test statistic (4.11), holds. Note that as m(g)i = 1 for all i = 1, 2, … , ng and g = 1, 0, the VIF reduces to 1 and thereby, test statistics (4.10) and (4.11) reduce to test statistics (2.3) and (2.4), respectively, for the case of no cluster sampling. Note also that simplifies to (1 + (m0(g) - 1) ρg) when m(g)i = m(g)0 for all i. Thus, we can use 1/(1 + (m0(g) − 1) ρg) to measure the relative efficiency of a statistic under cluster

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